Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\lbrace J_s \rbrace$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0 (X,L,J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr–Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton–Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove “independence of polarization” between Kähler quantizations and real polarizations. As an example, in the case of general flag varieties $X = G/B$ and for certain choices of highest weight $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V^{\ast}_\lambda$ to a collection of dirac delta functions supported at the Bohr–Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann–Berenstein–Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

中文翻译：

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极化、复曲面退化和 Newton-Okounkov 体的收敛

令$X$ 是维数$n$ 的光滑不可约复代数变体，$L$ 是$X$ 上的一个非常丰富的线丛。给定满足一些自然技术假设的 $(X,L)$ 的复曲面退化，我们在 $X$ 上构建复杂结构的变形 $\lbrace J_s \rbrace$ 并基于 $H 的 $\mathcal{B}_s$ ^0 (X,L,J_s)$ 使得 $J_0$ 是标准复数结构，并且在 $s \to \infty$ 的极限内，基本元素接近以玻尔-索末菲纤维为中心的狄拉克-德尔塔分布与 $X$ 及其环面退化相关的矩图。Newton-Okounkov 小体理论及其相关的复曲面退化表明，上述技术假设具有一定的普遍性。我们的结果显着概括了先前几何量化的结果，这证明了 Kähler 量化和实际极化之间的“极化独立性”。例如，在一般标志变体 $X = G/B$ 和最高权重 $\lambda$ 的某些选择的情况下，我们的结果在几何上构造了 $V^{\ ast}_\lambda$ 到 Bohr-Sommerfeld 纤维支持的狄拉克 delta 函数集合，这些函数与 Littelmann-Berenstein-Zelevinsky 弦多胞体的格点完全对应 $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$。